You can
put this solution on YOUR website!I'll do the first two to get you started
# 1
Start with the given system of equations:
Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.
So let's isolate y in the first equation
Start with the first equation
Subtract
from both sides
Rearrange the equation
---------------------
Since
, we can now replace each
in the second equation with
to solve for
Plug in
into the second equation. In other words, replace each
with
. Notice we've eliminated the
variables. So we now have a simple equation with one unknown.
Distribute the negative
Combine like terms on the left side
Add 7 to both sides
Combine like terms on the right side
Divide both sides by 7 to isolate x
Divide
-----------------First Answer------------------------------
So the first part of our answer is:
Since we know that
we can plug it into the equation
(remember we previously solved for
in the first equation).
Start with the equation where
was previously isolated.
Plug in
Multiply
Combine like terms
-----------------Second Answer------------------------------
So the second part of our answer is:
-----------------Summary------------------------------
So our answers are:
and
which form the point
)
Now let's graph the two equations (if you need help with graphing, check out this
solver)
From the graph, we can see that the two equations intersect at
. This visually verifies our answer.
graph of
(red) and
(green) and the intersection of the lines (blue circle).
# 2
Start with the given system of equations:
Multiply the both sides of the second equation by -2.
Distribute and multiply.
So we have the new system of equations:
Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:
Group like terms.
Combine like terms. Notice how the x terms cancel out.
Simplify.
Divide both sides by
to isolate
.
Reduce.
------------------------------------------------------------------
Now go back to the first equation.
Plug in
.
Multiply.
Subtract
from both sides.
Combine like terms on the right side.
Divide both sides by
to isolate
.
Reduce.
So our answer is
and
.
Which form the ordered pair
)
.
This means that the system is consistent and independent.
Notice when we graph the equations, we see that they intersect at
. So this visually verifies our answer.
Graph of
(red) and
(green)
